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Signal Characterization
Flow Measurement in Open Channels
Liquid Volume Measurement

Liquid volume measurement

A variety of technologies exist to measure the quantity of stored liquid in a vessel. Hydrostatic pressure, radar, ultrasonic, and tape-and-float are just a few of the more common technologies:



These liquid measuring technologies share a common trait: they infer the quantity of liquid in the vessel by measuring liquid height. If the vessel in question has a constant cross-sectional area throughout its working height (e.g. a vertical cylinder), then liquid height will directly correspond to liquid volume. However, if the vessel in question does not have a constant cross-sectional area throughout its height, the relationship between liquid height and liquid volume will not be linear.

For example, there is a world of difference between the height/volume functions for a vertical cylinder versus a horizontal cylinder:

The volume function for a vertical cylinder is a simple matter of geometry – height (h) multiplied by the cylinder’s cross-sectional area (πr2):

V = πr2h

Calculating the volume of a horizontal cylinder as a function of liquid height (h) is a far more complicated matter, because the cross-sectional area is also a function of height. For this, we need to apply calculus.

First, we begin with the mathematical definition of a circle, then graphically represent a partial area of that circle as a series of very thin rectangles:



In this sketch, I show the circle “filling” from left to right rather than from bottom to top. I have done this strictly out of mathematical convention, where the x (horizontal) axis is the independent variable. No matter how the circle gets filled, the relationship of area (A) to fill distance (h) will be the same.

If x2+y2 = r2 (the mathematical definition of a circle), then the area of each rectangular “slice” comprising the accumulated area between r and h r is equal to 2y dx. In other words, the total accumulated area between r and h r is:


Now, writing y in terms of r and x (y = r2 x2) and moving the constant “2” outside the integrand:


Consulting a table of integrals, we find this solution for the general form:


Applying this solution to our particular integral . . .


Knowing that the stored liquid volume in the horizontal tank will be this area multiplied by the constant length (L) of the tank, our formula for volume is as follows:

As you can see, the result is far from simple. Any instrumentation system tasked with the inference of stored liquid volume by measurement of liquid height in a horizontal cylinder must somehow apply this formula on a continuous basis. This is a prime example of how digital computer technology is essential to certain continuous measurement applications!

Spherical vessels, such as those used to store liquefied natural gas (LNG) and butane, present a similar challenge. The height/volume function is nonlinear because the cross-sectional area of the vessel changes with height.

Calculus provides a way for us to derive an equation solving for stored volume (V ) with height (h) as the independent variable. We begin in a similar manner to the last problem with the mathematical definition of a circle, except now we consider the filling of a sphere with a series of thin, circular disks:


If x2 + y2 = r2 (the mathematical definition of a circle), then the volume of each circular disk comprising the accumulated volume between r and h r is equal to πy2 dx. In other words, the total accumulated area between r and h r is:



Now, writing y in terms of r and x (y = r2 x2) and moving the constant π outside the integrand:


Immediately we see how the square and the square-root cancel one another, leaving us with a fairly simple integrand:


We may write this as the difference of two integrals:


Since r is a constant, the left-hand integral is simply πr2x. The right-hand integral is solvable by the power rule:


This function will “un-do” the inherent height/volume nonlinearity of a spherical vessel, allowing a height measurement to translate directly into a volume measurement. A “characterizing” function such as this is typically executed in a digital computer connected to the level sensor, or sometimes in a computer chip within the sensor device itself.

An interesting alternative to a formal equation for linearizing the level measurement signal is to use something called a multi-segment characterizer function, also implemented in a digital computer. This is an example of what mathematicians call a piecewise function: a function made up of line segments. Multi-segment characterizer functions may be programmed to emulate virtually any continuous function, with reasonable accuracy:



The computer correlates the input signal (height measurement, h) to a point on this piecewise function, linearly interpolating between the nearest pair of programmed coordinate points. The number of points available for multi-point characterizers varies between ten and one hundred2 depending on the desired accuracy and the available computing power.

2There is no theoretical limit to the number of points in a digital computer’s characterizer function given sufficient processing power and memory. There is, however, a limit to the patience of the human programmer who must encode all the necessary x, y data points defining this function. Most of the piecewise characterizing functions I have seen available in digital instrumentation systems provide 10 to 20 (x, y) coordinate points to define the function. Fewer than 10 coordinate points risks excessive interpolation errors, and more than 20 would just be tedious to set up.

Although true fans of math might blanch at the idea of approximating an inverse function for level measurement using a piecewise approach rather than simply implementing the correct continuous function, the multi-point characterizer technique does have certain practical advantages. For one, it is readily adaptable to any shape of vessel, no matter how strange. Take for instance this vessel, made of separate cylindrical sections welded together:


Here, the vessel’s very own height/volume function is fundamentally piecewise, and so nothing but a piecewise characterizing function could possibly linearize the level measurement into a volume measurement!

Consider also the case of a spherical vessel with odd-shaped objects welded to the vessel walls, and/or inserted into the vessel’s interior:



The volumetric space occupied by these structures will introduce all kinds of discontinuities into the transfer function, and so once again we have a case where a continuous characterizing function cannot properly linearize the level signal into a volume measurement. Here, only a piecewise function will suffice.

To best generate the coordinate points for a proper multi-point characterizer function, one must collect data on the storage vessel in the form of a strapping table. This entails emptying the vessel completely, then filling it with measured quantities of liquid, one sample at a time, and taking level readings:

  Introduced liquid volume     Measured liquid level 
150 gallons  2.46 feet
300 gallons  4.72 feet
450 gallons 5.8 feet
600 gallons (etc., etc.)
750 gallons (etc., etc.)

Each of these paired numbers would constitute the coordinates to be programmed into the characterizer function computer by the instrument technician or engineer:

Many “smart” level transmitter instruments possess enough computational power to perform the level-to-volume characterization directly, so as to transmit a signal corresponding directly to liquid volume rather than just liquid level. This eliminates the need for an external “level computer” to perform the necessary characterization. The following screenshot was taken from a personal computer running configuration software for a radar level transmitter3, showing the strapping table data point fields where a technician or engineer would program the vessel’s level-versus-volume piecewise function:



This configuration window actually shows more than just a strapping table. It also shows the option of calculating volume for different vessel shapes (vertical cylinder is the option selected here) including horizontal cylinder and sphere. In order to use the strapping table option, the user would have to select “Strapping Table” from the list of Tank Types. Otherwise, the level transmitter’s computer will attempt to calculate volume from an ideal tank shape.

3The configuration software is Emerson’s AMS, running on an engineering workstation in a DeltaV control system network. The radar level transmitter is a Rosemount model 3301 (guided-wave) unit.

Radiative temperature measurement

Temperature measurement devices may be classified into two broad types: contact and non-contact. Contact-type temperature sensors detect temperature by directly touching the material to be measured, and there are several varieties in this category. Non-contact temperature sensors work by detecting light emitted by hot objects.

Energy radiated in the form of electromagnetic waves (photons, or light) relates to object temperature by an equation known as the Stefan-Boltzmann equation, which tells us the rate of heat lost by radiant emission from a hot object is proportional to the fourth power of its absolute temperature:

P = eσAT4


  P = Radiated energy power (watts)

  e = Emissivity factor (unitless)

  σ = Stefan-Boltzmann constant (5.67 × 10-8 W / m2 K4)

  A = Surface area (square meters)

  T = Absolute temperature (Kelvin)


Solving for temperature (T) involves the use of the fourth root, to “un-do” the fourth power function inherent to the original function:

Any optical temperature sensor measuring the emitted power (P) must “characterize” the power measurement using the above equation to arrive at an inferred temperature. This characterization is typically performed inside the temperature sensor by a microcomputer.

Analytical measurements

A great many chemical composition measurements may be made indirectly by means of electricity, if those measurements are related to the concentration of ions (electrically charged molecules). Such measurements include:


  • pH of an aqueous solution

  • Oxygen concentration in air

  • Ammonia concentration in air

  • Lead concentration in water


The basic principle works like this: two different chemical samples are placed in close proximity to each other, separated only by an ion-selective membrane able to pass the ion of interest. As the ion activity attempts to reach equilibrium through the membrane, an electrical voltage is produced across that membrane. If we measure the voltage produced, we can infer the relative activity of the ions on either side of the membrane.

Not surprisingly, the function relating ion activity to the voltage generated is nonlinear. The standard equation describing the relationship between ionic activity on both sides of the membrane and the voltage produced is called the Nernst equation:



  V = Electrical voltage produced across membrane due to ion exchange (volts)

  R = Universal gas constant (8.315 J/molK)

  T = Absolute temperature (Kelvin)

  n = Number of electrons transferred per ion exchanged (unitless)

  F = Faraday constant (96,485 coulombs per mole)

  a1 = Activity of ion in measured sample

  a2 = Activity of ion in reference sample (on other side of membrane)


A practical application for this technology is in the measurement of oxygen concentration in the flue gas of a large industrial burner, such as what might be used to heat up water to generate steam. The measurement of oxygen concentration in the exhaust of a combustion heater (or boiler) is very important both for maximizing fuel efficiency and for minimizing pollution (specifically, the production of NOx molecules). Ideally, a burner’s exhaust gas will contain no oxygen, having consumed it all in the process of combustion with a perfect stoichiometric mix of fuel and air. In practice, the exhaust gas of an efficiently-controlled burner will be somewhere near 2%, as opposed to the normal 21% of ambient air.

One way to measure the oxygen content of hot exhaust is to use a high-temperature zirconium oxide detector. This detector is made of a “sandwich” of platinum electrodes on either side of a solid zirconium oxide electrolyte. One side of this electrochemical cell is exposed to the exhaust gas (process), while the other side is exposed to heated air which serves as a reference:



The electrical voltage generated by this “sandwich” of zirconium and platinum is sent to an electronic amplifier circuit, and then to a microcomputer which applies an inverse function to the measured voltage in order to arrive at an inferred measurement for oxygen concentration. This type of chemical analysis is called potentiometric, since it measures (“metric”) based on an electrical voltage (“potential”).

The Nernst equation is an interesting one to unravel, to solve for ion activity in the sample (a1) given voltage (V ):



Multiplying both sides by nF:


Dividing both sides by RT:


Applying the rule that the difference of logs is equal to the log of the quotient:


Adding ln a2 to both sides:


Making both sides of the equation a power of e:


Canceling the natural log and exponential functions on the right-hand side:


In most cases, the ionic activity of
a2 will be relatively constant, and so ln a2 will be relatively constant as well. With this in mind, we may simplify the equation further, using k as our constant value:


Substituting k for ln a2:


Applying the rule that the sum of exponents is the product of powers:


If k is constant, then ek will be constant as well (calling the new constant C):


Analytical instruments based on potentiometry must evaluate this inverse function to “undo” the Nernst equation to arrive at an inferred measurement of ion activity in the sample given the small voltage produced by the sensing membrane. These instruments typically have temperature sensors as well built in to the sensing membrane assembly, since it is apparent that temperature (T) also plays a role in the generation of this voltage. Once again, this mathematical function is typically evaluated in a microprocessor.




Lipt´ak, B´ela G., Instrument Engineers’ Handbook – Process Measurement and Analysis Volume I, Fourth Edition, CRC Press, New York, NY, 2003.

Stewart, James, Calculus: Concepts and Contexts, 2nd Edition, Brooks/Cole, Pacific Grove, CA, 2001.

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